A simple proof of a duality theorem with applications in scalar and anisotropic viscoelasticity
aa r X i v : . [ m a t h - ph ] M a y Keywords: viscoelasticity,completely monotone, Bernstein, complete Bern-stein, Stieltjes function
Abstract
A new concise proof is given of a duality theorem connecting com-pletely monotone relaxation functions with Bernstein class creep functionsin one-dimensional and anisotropic 3D viscoelasticity. The proof makesuse of the theory of complete Bernstein functions and Stieltjes functionsand is based on a relation between these two function classes. simple proof of a duality theorem withapplications in scalar and anisotropicviscoelasticity Andrzej Hanygaul. Bitwy Warszawskiej 1920 r 14/5202-366 Warszawa, [email protected]
Notations [0 , ∞ [:= { x ∈ R | ≤ x < ∞} ]0 , ∞ [:= { x ∈ R | < x < ∞} D f ( t ) := d f ( t ) / d t = ˙ f ( t ) It happens so that all the mathematical models of viscoelastic relaxation moduliare completely monotone (CM) functions. This is partly due to the interpolationprocedures applied in deriving the relaxation moduli (such as Prony sums), or,more often, the creep functions from discrete experimental data. The set of CMfunctions is a very thin subset in the spaces of continuous or smooth fuctions,i. e. in an arbitrarily small neighborhood of a CM function in the C topologythere are functions which are not CM, but the interpolation procedures arebased a priori on CM functions such as the exponential function exp( − kt ). Bythe duality theorem considered below the corresponding creep functions areBernstein functions (BFs).The assumption of the CM property of relaxation moduli has very power-ful consequences especially for viscoelastic wave propagation [5, 7]. Thereforeit is convenient to stick to this assumption so long as it does not contradictexperimental data.In this paper I shall demonstrate the application of the complete Bernsteinfunctions and the Stieltjes functions in the proof of a well known duality relationbetween locally integrable completely monotone (LICM) relaxation moduli R and Bernstein class creep functions C . I recall the viscoelastic duality relation21, 2, 3] ˜ R ( p ) ˜ C ( p ) = p − (1)where ˜ f ( p ) := Z ∞ e − pt f ( t ) d t (2)denotes the Laplace transform for any locally integrable function f such thatthe transform exists.Experimental evidence suggests that the relaxation moduli of all the knownviscoelastic media are completely monotone functions of time. It was previouslyshown by other methods [1, 2, 4] that, apart from some singular terms, if R islocally integrable completely monotone (LICM), then the creep function C isa Bernstein function and conversely. The new proof of the same relation ismore elegant and concise than the previously given proofs [1, 2, 4] of the samerelation. It is based on the theory of complete Bernstein functions and Stieltjesfunctions [6].The main advantage of this approach is availability of integral representa-tions of Stieltjes and complete Bernstein functions. As soon as we have estab-lished that a function of interest is a Stieltjes or complete Bernstein functionwe have a decomposition of that function in the form of the sum of three termswhich is complete. For example we discover that for a given creep function thedual relaxation function must contain a Newtonian viscosity term.In [4] the results of [2] were extended to tensor-valued relaxation moduli andcreep functions. In this paper we shall also study the duality relation for tensor-valued relaxation moduli and creep functions. The tensor-valued functions willbe treated as matrix-valued functions on R . We shall use the theory of matrix-valued complete Bernstein and Stieltjes functions developed in [7].While in the scalar case we obtain a fairly complete set of relations betweenrelaxation and creep, in the anisotropic case there are some limitations due tothe complex questions of matrix invertibility.These relations are of paramount importance for deducing the relaxationmodulus from creep test data. It is convenient for our considerations to consider functions f in L ([0 , ∞ [) asconvolution operators f ∗ mapping g ∈ L ([0 , ∞ [) to f ∗ g ∈ L [0 , ∞ [).The convolution of two locally integrable functions f and g on [0 , ∞ [ isdefined by the formula ( f ∗ g )( t ) := Z t f ( s ) g ( t − s ) d s (3)If the Laplace transforms ˜ f ( p ) and ˜ g ( p ) exist for some p ≥
0, then the convolu-tion f ∗ g also has the Laplace transform at p and( f ∗ g ) e ( p ) = ˜ f ( p ) ˜ g ( p ) (4)3e shall also need an identity operator on L ([0 , ∞ [):U f = f (5)For the sake of convenience we shall also write (5) in the form u ∗ f = f (6)Extending (4) to (5) we have ˜ u ( p ) ˜ f ( p ) = ˜ f ( p ), whence˜ u ( p ) = 1 , p ≥ σ = R ∗ ˙ ǫ , where σ represents the stress, ˙ ǫ is the strain rateand the relaxation modulus R = N u + f , where f ∈ L ([0 , ∞ [), then σ = N ˙ ǫ + f ∗ ˙ ǫ . In the absence of the second term the first term represents Newtonianviscosity. We shall however see that for the validity of the duality relation theappearance of a term b u is necessary.The mathematical function classes required in the proof of the duality the-orem are explained in the appendix along with their properties which will beneeded in the following. Theorem 1 If f is LICM and not identically zero and f ( t ) = f ( t ) + β u ( t ) (8) where β ≥ , then / [ p ˜ f ( p )] = p ˜ h ( p ) (9) where h is a Bernstein function.We also have < f (0) ≤ ∞ . If β > or f ( t ) is unbounded at 0 then h (0) = 0 , otherwise h (0) = 1 /f (0) .The limit f ∞ := lim t →∞ f ( t ) exists and is non-negative.If f ∞ > , then for t → ∞ the function h ( t ) tends to /f ∞ , otherwise it divergesto infinity. Proof.
We shall use a few theorems on of the CBFs and Stieltjes functions quotedin the appendix to construct the Bernstein function h .Since f is LICM, there is a non-negative real number a and Borel measure µ on ]0 , ∞ [ satisfying the inequality Z ]0 , ∞ [ (1 + s ) − µ (d s ) < ∞ (10)such that f ( t ) = a + Z ]0 , ∞ [ e − st µ (d s ) (11)4t follows that ˜ f ( p ) = a/p + Z ]0 , ∞ [ ( s + p ) − µ (d s ) (12)hence, by eq. (31), p ˜ f ( p ) = β p + p ˜ f ( p ) is a CBF and 1 / [ p ˜ f ( p )] is a Stieltjesfunction. Hence there are non-negative real numbers a, b and a Borel measure ν on ]0 , ∞ [ satisfying the inequality Z ]0 , ∞ [ (1 + s ) − ν (d s ) < ∞ (13)such that 1 / [ p ˜ f ( p )] = a + bp + Z ]0 , ∞ [ ν (d r ) r + p (14)The last term is the Laplace transform ˜ g ( p ) of the LICM function g ( t ) := Z ]0 , ∞ [ e − tr ν (d r ) (15)If G ( t ) := Z t g ( s ) d s + a then p ˜ G ( p ) = ˜ g ( p ) + a and G is a Bernstein function.Concerning the term b/p appearing in (14) we note that b/p is the Laplacetransform of the Bernstein function b t, t ≥
0. Hence eq. (9) holds with theBernstein function h ( t ) := b t + G ( t ) (16)It remains to consider the limits of these functions.The function f ( t ) ≥ f f ( t ) may be un-bounded at 0, otherwise the limit f (0) of f ( t ) at 0 exists and f (0) >
0. Onthe other hand h (0) ≥ h (0) = lim p →∞ / [ β p + p e f ( p )]If β > f ( t ) is unbounded at 0, then h (0) = 0, otherwise h (0) = 1 /f (0).The function f satisfies the inequalities 0 ≤ f ( t ) ≤ f (1) for t > f ∞ := lim t →∞ f ( t ) exists and isnon-negative. We now note thatlim p → [ p ˜ f ( p )] = lim p → [ p e f ( p ))] = f ∞ , If f ∞ > t →∞ h ( t ) = lim p → [ p ˜ h ( p )] = 1 /f ∞
5n account of (9). Otherwise h ( t ) is unbounded at infinity and the limitlim t →∞ h ( t ) does not exist. (cid:3) Theorem 2 If h is a B not identically zero, then there is a LICM function f and a real b ≥ such that / [ p ˜ h ( p )] = p ˜ f ( p ) (17) where f ( t ) := b u ( t ) + f ( t ) (18) The function h ( t ) is either bounded and tends to a positive limit at infinityor it diverges to infinity. In the first case we have the identity lim t →∞ f ( t ) = lim t →∞ /h ( t ) , otherwise f ( t ) tends to 0 at infinity.If h (0) > , then f ( t ) is bounded at 0, f (0) = 1 /h (0) and b = 0 , otherwiseeither b > or f ( t ) is unbounded at 0.If h (0) = 0 , then b = 1 /h ′ (0) for h ′ (0) ≥ . Proof.
Since p ˜ h ( p ) = [ h ′ ]˜( p ) + h (0) and the derivative h ′ of h is LICM, p ˜ h ( p ) is aStieltjes function and therefore 1 / [ p ˜ h ( p )] is a CBF.Eq. (31) implies that there are two reals a, b ≥ ν satisfying (13) such that1 / [ p ˜ h ( p )] = a + b p + p Z ]0 , ∞ [ ν (d r ) r + p Let f ( t ) := a + Z ]0 , ∞ [ e − rt ν (d r )and f ( t ) := f ( t ) + b u ( t ) f is clearly LICM and eq. (17) is satisfied.Furthermore, lim t →∞ h ( t ) > t →∞ /h ( t ) = lim p → / [ p ˜ h ( p )] = lim p → [ p ˜ f ( p )] = lim p → e f ( p ) = lim t →∞ f ( t ) (19)At the other end we note that h (0) ≥ b = 0, and f ( t ) is boundedat 0, then h (0) = lim p →∞ [ p ˜ h ( p )] = lim p →∞ / [ b p + p e f ( p )] = 1 /f (0) , h (0) = 0.Hence if h (0) >
0, then b = 0 and f ( t ) tends to 1 /h (0) for t →
0, while if h (0) = 0 then either b > f ( t ) diverges to infinity at 0.If h (0) = 0, then lim p →∞ [ p ˜ h ( p )] = lim p →∞ [ p e h ′ ( p )] = lim t → h ′ ( t ). Thelast limit exists because h ′ is LICM, but it may be infinite. On the other hand(9) implies that lim p →∞ [ p ˜ h ( p )] = lim p →∞ / [ b + e f ( p )]. We now note thatlim p →∞ e f ( p ) = lim p →∞ { p [1 /p e f ( p )] } = lim t → Z t f ( t ) d t = 0 (20)The last equation in (20) follows from the fact that f is integrable over[0 , t ≤ Z t f ( t ) d t = Z f ( s ) θ ( t − s ) d s where θ denotes the unit step function, f ( s ) θ ( t − s ) ≤ f ( s ) and f ( s ) θ ( t − s ) → ≤ s ≤ t →
0, hence the last equation in (20) follows from theLebesgue dominated convergence theorem.We thus conclude that in the case of h (0) = 0 b = 1 / lim t → h ′ ( t ) . with b = 0 if h ′ ( t ) → ∞ for t → (cid:3) Constitutive equations of anisotropic viscoelastic media in three-dimensionalspace assume the following formΣ I = X J =1 R IJ ∗ ˙ E J , I = 1 , . . . E I = X J =1 C IJ ∗ ˙Σ J , I = 1 , . . . I = 1 , , I = 4 , , , , E k = ǫ kk for k = 1 , , E l = 2 / ǫ mn for l = m, n, and m = n ,with similar rules for R IJ . While R ijkl ( t ), C ijkl ( t ) and ǫ ij are tensor-valuedfunctions, the corresponding 6-dimensional objects are 6 × × R ijkl = R klij , C ijkl = C klij the functions R IJ ( t ), C IJ ( t ) defined on ]0 , ∞ [ take values in the space S + of positive semi-definite symmetric matrices 6 × × R ( t ) and C ( t ), respectively.7e shall study the relation p ˜ R ( p ) = p − ˜ C ( p ) − (23)[4]. Matrix-valued CB and Stieltjes functions were studied in [7]. Some resultsrelevant for us are collected in Appendix B. There is a close analogy betweenthem and the results of Appendix A used in the previous section.Recall that R ( t ) at 0 and C ( t ) at infinity may be unbounded. If however v T R ( t ) v is bounded for every v ∈ R , then it tends to a limit. By polarizationwe conclude that R ( t ) tends to a limit which we denote as the value R (0). Simi-larly, if the creep function v T C v is bounded for each v ∈ R , then lim t →∞ C ( t )exists. Theorem 3 If R ( t ) = u ( t ) N + F ( t ) , where F is a S + -valued LICM and N ∈ S + and ( ∗ ) for each non-zero vector v ∈ R the function R IJ ( t ) v I v J is not identicallyzero,then there is a S + -valued Bernstein function C such that equation (23) holds.The limit lim t → C ( t ) = C (0) always exists and is positive semi-definite.If N > , then C (0) = 0 and lim t → C ′ ( t ) = N − .If N = 0 , the limit lim t → F ( t ) exists and is invertible, then C (0) = [lim t → F ( t )] − .The limit lim t →∞ R ( t ) =: F ∞ always exists and is positive-semi-definite.If F ∞ is invertible, then lim t →∞ C ( t ) exists and lim t →∞ C ( t ) = h lim t →∞ R ( t ) i − (24) Proof.
On account of (37) p ˜ R ( p ) = p N + B + p Z ]0 , ∞ [ ( p + r ) − G ( r ) µ (d r ) , hence p ˜ R ( p ) is an S + -valued CBF.On account of ( ∗ ) the matrix ˜ R ( p ) has an inverse for p >
0. Indeed, forevery vector v there is a positive number t ∗ ( v ) such that v T R ( t ∗ ) v >
0, while v T R ( t ) v ≥ t >
0. Hence ˜ R ( p ) > p ≥ R ( p ) has aninverse for p ≥ p ˜ R ( p is thus an S + -valued Stieltjes function and has the form A + p − D + Z ]0 , ∞ [ ( p + r ) − H ( r ) ν (d r ) , (25)where A , D ∈ S + , ν is a Borel measure on ]0 , ∞ [ satisfying (13) and H is abounded measurable S + -valued function defined ν -almost everywhere on ]0 , ∞ [.On account of equation (23) p ˜ C ( p ) has the form given by equation (25). Itfollows that C ( t ) = A + t D + Z t K ( s ) d s, K ( t ) := Z ]0 , ∞ [ e − rt H ( r ) ν (d r )is a LICM. It follows that C is an S + -valued Bernstein function.We now turn to the limits of the creep function.Note that the limit F ∞ := lim t →∞ F ( t ) always exists and is positive semi-definite. Indeed, the function v T F ( t ) v is a non-negative non-increasing forevery v ∈ R , hence it tends to a limit. By the polarization argument v T F ( t ) w also tends to a limit for t → ∞ for every v , w . Hence lim t →∞ F ( t ) exists and ispositive semi-definite.By an analogous argument C (0) is defined and positive semi-definite.Nowlim t → C ( t ) = lim p →∞ [ p ˜ C ( p )] = lim p →∞ " p N + B + p Z ]0 , ∞ [ ( r + p ) − G ( r ) µ (d r ) − If N >
0, then the right-hand side can be recast in the formlim p →∞ p − " N + p − B + Z ]0 , ∞ [ ( r + p ) − G ( r ) µ (d r ) − The last term is the Laplace transform of the integral R t F ( s ) d s , where F ( t ) := R ]0 , ∞ [ e − rt G ( r ) µ (d r ) is a LICM function. Hence the limit at p → ∞ of thatterm is equal to lim t → Z t F ( s ) d s, which vanishes because F ( t ) is integrable over [0 , p →∞ " N + p − B + Z ]0 , ∞ [ ( r + p ) − G ( r ) µ (d r ) = N and is invertible. Therefore lim p →∞ h N + p − B + R ]0 , ∞ [ ( r + p ) − G ( r ) µ (d r ) i − exists and equals N − . It follows that lim t → C ( t ) = 0 in this case.Furthermore, under the same assumptionlim t → C ′ ( t ) = lim p →∞ [ p ˜ C ( p )] = lim p →∞ " N + p − B + Z ]0 , ∞ [ ( r + p ) − G ( r ) µ (d r ) − = N − If N = 0, thenlim t → C ( t ) = lim p →∞ " B + p Z ]0 , ∞ [ ( r + p ) − G ( r ) µ (d r ) − p → ∞ is lim t → F ( t ), which may be infinite.However, if the last limit is finite and B + lim t → F ( t ) = R (0) − is invertible,then C (0) = [ R (0)] − .Finally,lim t →∞ C ( t ) = lim p → [ p ˜ C ( p )] = lim p → " p N + B + p Z ]0 , ∞ [ ( p + r ) − G ( r ) µ (d r ) − If the limit lim p → h B + p R ]0 , ∞ [ ( p + r ) − G ( r ) µ (d r ) i ≡ B + lim t →∞ F ( t ) ≡ lim t →∞ R ( t ) exists and is invertible, then equation (24) is satisfied. (cid:3) Theorem 4 If C is a S + -valued Bernstein function, and ( ∗∗ ) for each non-zero vector v ∈ R the function R IJ ( t ) v I v J is not identicallyzero,then there is an S + -valued LICM function F and N ∈ S + such that R ( t ) = u ( t ) N + F ( t ) (26) satisfies equation (23) .The creep modulus has the form C ( t ) = A + t B + Z t Q ( s ) d s, (27) where A , B ∈ S + , lim t →∞ C ′ ( t ) = B and Q is a LICM.If A ≡ C (0) has an inverse, then lim t → R ( t ) = C (0) − (28) If B > , then lim t →∞ R ( t ) = 0 .If B = 0 , the limit of C ( t ) at infinity exists and is invertible, then lim t →∞ R ( t ) = h lim t →∞ C ( t ) i − (29) Proof.
Condition ( ∗∗ ) ensures that the matrix ˜ C ( p ) is invertible for p ≥ C ( t ) = R t L ( s ) d s , where L is an S + -valued LICM func-tion. Consequently p ˜ C ( p ) = ˜ L ( p ) is an S + -valued Stieltjes function. Its inverseis an S + -valued CBF and it has the form N + p B + p Z ]0 , ∞ [ ( p + r ) − G ( r ) µ (d r ) , with N , B ∈ S + , a Borel measure µ satisfying (10) and a bounded measurable S + -valued function G on ]0 , ∞ [. Equation (23) is satisfied if R is given byequation (26) with F ( t ) := B + Z ]0 , ∞ [ e − rt G ( r ) µ (d r ) . v ∈ R the function v T R ( t ) v is CM, it has a non-negativelimit at infinity. Hence lim t →∞ R ( t ) exists and is positive-semidefinite.Since p ˜ C ( p ) is a Stieltjes function p ˜ C ( p ) = A + p − B + Z ]0 , ∞ [ ( p + r ) − H ( r ) ν (d r )for some A , B ∈ S + , a Borel measure ν satisfying (10) and a bounded S + -valuedfunction H . Defining a LICM function Q ( t ) := R ]0 , ∞ [ e − rt H ( r ) ν (d r ) we obtainequation (27). C ( t ) = A + t B + R t Q ( s ) d s .Consider lim t →∞ Q ( t ) = lim t →∞ Z ]0 , ∞ [ e − rt H ( r ) ν (d r ) . On account of the inequality e − x ≤ (1 + x ) − for x > r ) − ν (d r ) for t >
1. On account of (10) and theLebesgue Dominated Convergence Theorem lim t →∞ Q ( t ) = 0. Consequentlythe derivative C ′ ( t ) tends to B at infinity.If B >
0, then the function C ( t ) diverges at infinity.We now investigate the limit lim t → R ( t ) = lim p →∞ h A + p − B + ˜ Q ( p ) i − .Noting that lim p →∞ ˜ Q ( p ) = lim t → R t Q ( s ) d s = 0 (because Q is locally inte-grable) we get lim t → R ( t ) = A − ≡ C (0) − if C (0) is invertible..We now note thatlim t →∞ R ( t ) = lim p → (cid:26) p h p A + B + p ˜ Q ( p ) i − (cid:27) . The last term in the square brackets tends to lim t → Q ( t ) = 0.If B >
0, then the expression in the square brackets tends to B , and thereforelim t →∞ R ( t ) = 0.If B = 0, then lim t →∞ R ( t ) = lim p → h A + ˜ Q ( p ) i − . The limit of theexpression in the square brackets is A + R ∞ Q ( s ) d s ) = lim t →∞ C ( t ). If the lastlimit exists and is invertible, then equation (29) is satisfied. (cid:3) Condition ( ∗ ) ensures that a non-zero strain always causes a non-zero stress.Condition ( ∗∗ ) ensures that a non-zero stress always causes a non-zero strain. We have identified the Laplace transforms of the relaxation modulus and creepfunction as members of appropriate classes of functions. Availability of integralrepresentations for these classes allows a complete listing of the component terms11elevant for the duality equation. We have thus determined all the componentsof the relaxation modulus and the creep function in a class of viscoelastic modelscharacterized by completely monotone relaxation.The proofs of equation (1) suggest that the relaxation modulus and the creepfunction should be considered as convolution operators. Such an approach allowsincorporating the identity operator in this class. The identity operator cannotbe ignored in the context of the duality equation (1). Even though one mightset the Newtonian viscosity term β = 0 in (8) or N = 0 in equation (26), a”Newtonian viscosity” term b t or t B will pop up in equation (16).Given an anisotropic structure of the medium ane might construct the func-tion F ( t ) in Theorem 3 by setting G ( r ) = P J =1 λ J ( r ) S J ⊗ S J with 0 ≤ λ J ( r ) ≤ (1), for J = 1 , . . . S J held constant. For F we get thefollowing formula F ( t ) = B + X J =1 f J ( t ) S J ⊗ S J with f J ( t ) = Z ]0 , ∞ [ e − rt λ J ( r ) µ (d r )a LICM function. f I ( t ) represents the relaxation of the eigenstress Σ I scausedby a jump of the eigenstrain E I from 0 to E I .In this case the undrlying anisotropic directional structure is not affectedby relaxation and the functions f J account for different relaxations of differenteigenstresses. The eigenstrains are determined by the anisotropy class. A A few relevant properties of LICM, completeBernstein and Stieltjes functions.
For details, see [6, 8].In order to focus on those statements which are of use for us we shall consideras definitions some statements that appear as theorems in the references citedabove.An infinitely differentiable function f is said to be LICM if it is completelymonotone: ( − n D n f ( t ) ≥ n = 0 , , , ... and integrable in a neighborhood of 0 [8].If f ( t ) is LICM, then there is a real number a ≥ µ on ]0 , ∞ [ satisfying (10) such that f ( t ) = a + Z ]0 , ∞ [ e − rt µ (d r )The last equation can also be recast in a more familiar form f ( t ) = Z [0 , ∞ [ e − rt µ (d r )12y defining µ ( { } ) = a .The inverse implication is also true.A Bernstein function is defined as an infinitely differentiable function g on[0 , ∞ [ satisfying the inequalities g ( t ) ≥ − n D n g ( t ) ≤ n = 1 , , . . . .If f is LICM, a, b ≥
0, then g ( t ) = R t g ( s ) d s + a + b t is a Bernstein function.The derivative of a Bernstein function is LICM.A function f is a Stieltjes function if there are two real numbers a, b ≥ µ satisfying (10) such that f ( p ) = a + bp + Z ]0 , ∞ [ µ (d r ) p + r (30)The integration extends over 0 < r < ∞ . The second term can be incorporatedin the integral by extending the integration to [0 , ∞ [, but we prefer to keep itseparate [6], Def. 2.1.A function f is a CBF if there are two real numbers a, b ≥ ν on ]0 , ∞ [ satisfying (13) such that f ( p ) = a + b p + p Z ]0 , ∞ [ ν (d r ) p + r (31)[6], Thm 6.2. f ( p ) is a Stieltjes function if and only if p f ( p ) is a CBF. This statementfollows from the integral representations of CBFs and Stieltjes functions above.The following non-linear relation between the CBFs and the Stieltjes func-tions is the key to the proof above: A function f ( p ) not identically zero is aStieltjes function if and only if 1 /f ( p ) is a CBF not identically 0. [6], Thm 7.3. B Some relevant properties of matrix-valued func-tions of the Stieltjes and complete Bernsteinclass.
For details see [7].Let S + denote the set of non-negative symmetric matrices.A symmetric matrix-valued function A ( t ) is CM if( − n D n A ( t ) ≥ n = 0 , , . . . where B ≥ v T M v ≥ v ∈ R .The function A ( t ) is LICM if it is CM and locally integrable.If A is LICM then for every vector v ∈ R the function v T A v is LICM. ByBernstein’s theorem there is a Borel measure m v on [0 , ∞ [ such that v T A ( t ) v = Z [0 . ∞ [ e − rt m v (d r )13nd Z ]0 . ∞ [ (1 + r ) − m v (d r ) < ∞ (32)We now note that 4 v T A ( t ) w = Z [0 , ∞ [ e − rt M v , w (d r ) (33)for every v , w ∈ R , where M v , w ( E ) := m v + w ( E ) − m v − w ( E ) for every Borel E ⊂ [0 , ∞ [ and v , w ∈ R . From equation (33), using the uniqueness of theLaplace transform, follows that M λ v + z , w ( E ) = λ M v , w ( E ) + M z , w ( E ) hencethere is a matrix M ( E ) such that M v , w ( E ) = v T M ( E ) w . This matrix isalso symmetric. Since m v ( E ) ≥ m ( E ) = 0, the matrix E is positivesemi-definite.Furthermore, denoting M ( E ) =: H , we note that a square root H / suchthat H = H / H / (see, e. g. [7] for the definition). The square root is apositive semi-definite symmetric matrix, hence | v T H w | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) H / v (cid:17) T (cid:16) H / w (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ [ v T H v ] [ w T H w ] (34)Since H ≥
0, the right-hand side of (34) is bounded from above by trace( H ) | v | w | .Define the Borel measure µ ( E ) := trace( M ( E )) for Borel E ⊂ [0 , ∞ [. Since | v T M ( E ) w | ≤ µ ( E ) | v | | w | , by the Radon-Nikodym theorem there is a boundedmeasurable function g v , w on [0 , ∞ [, defined µ -almost everywhere, such that M ( E ) = R E g v , w ( r ) µ (d r ) for every Borel E ⊂ ]0 , ∞ [. Repeating an argumentused above one can prove that g v , w ( r ) = v T G ( r ) w , where G ( r ) is an S + -valued function on [0 , ∞ [. We conclude that for every symmetric LICM function A there is a Borel measure µ and a bounded S + -valued function G on [0 , ∞ [such that A ( t ) = Z [0 , ∞ [ e − rt G ( r ) µ (d r ) (35)We note that on account of (32) the Borel measure µ satisfies equation (10).If µ ( { } ) >
0, then G (0) is defined and (35) can be recast in the form A ( t ) = B + Z ]0 , ∞ [ e − rt G ( r ) µ (d r ) (36)where B := µ ( { } ) G (0) is a positive semi-definite symmetric matrix. If µ ( { } ) =0 then B = 0.Calculation of limits of A ( t ) imposes the necessity to split A ( t ) into twoterm and consider µ as a Borel measure on ]0 , ∞ [.The Laplace transform of the S + -valued LICM A ( t ) is given by the equation˜ A ( p ) = p − B + Z ]0 , ∞ [ ( p + r ) − G ( r ) µ (d r ) (37)14n S + -valued Bernstein function is an indefinite integral of a S + -valuedLICM function.We shall now recall some results from Appendix B of [7].A matrix-valued Stieltjes function Y ( p ) has the following integral represen-tation: Y ( p ) = B + p − C + Z ]0 , ∞ [ ( p + r ) − G ( r ) µ (d r ) (38)where B , C ∈ S + , µ is a Borel measure on ]0 , ∞ [ satisfying (10) and G ( r ) isan S + -valued function defined µ -almost everywhere on ]0 , ∞ [. Conversely, anymatrix-valued function with the integral representation (38) is an S + -valuedStieltjes function.An S + -valued CBF Z ( p ) has the following integral representation: Z ( p ) = B + p C + p Z ]0 , ∞ [ ( p + r ) − H ( r ) ν (d r ) (39)where B , C ∈ S + , ν is a Borel measure on ]0 , ∞ [ satisfying (13) and H ( r )is an S + -valued function defined ν -almost everywhere on ]0 , ∞ [. Conversely,any matrix-valued function with the integral representation (39) is a S + -valuedCBF.It follows immediately that the the function p − Z ( p ), where Z is an S + -valued CBF function, is an S + -valued Stieltjes function.According to Lemma 3 op. cit. if Z ( p ) is an invertible S + -valued CBF then Z ( p ) − is an S + -valued Stieltjes function and conversely. References [1] Molinari A. 1973 Sur la relation entre fluage et relaxation en visco´elasticit´enon-lin´eaire.
C. R. Acad. Sci. Paris S´er. A , 621–623.[2] Hanyga A. 2005 Physically acceptable viscoelastic models. In Hutter K,Wang Y, editors,
Trends in Applications of Mathematics to Mechanics pp.125–136. Shaker Verlag GmbH.[3] Mainardi F. 2010
Fractional Calculus and Waves in Viscoelasticity . WorldScientific.[4] Hanyga A, Seredy´nska M. 2007 Relations between relaxation modulus andcreep compliance in anisotropic linear viscoelasticity.
J. of Elasticity ,41–61.[5] Hanyga, A.: Wave propagation in linear viscoelastic media with completelymonotonic relaxation moduli. Wave Motion , 909–928 (2013)[6] Schilling RL, Song R, Vondraˇcek Z. 2010 Bernstein Functions. Theory andApplications . Berlin: De Gruyter.157] Hanyga A. 2016 Wave propgation in anisotropic elasticity.
J. of Elasticity , 231–254.[8] Seredy´nska M, Hanyga A. 2010 Relaxation, dispersion, attenuation andfinite speed of propagation speed in viscoelastic media.
J. Math. Phys.51